Let $X_1,...,X_n$ be i.i.d. standard normal random variables. How to show that there is constant $c>0$ such that for every $a_k>0$ and for every $n>0$: $P(\sum_{k=1}^{n}a_kX_k^2>\sum_{k=1}^{n}a_k+c\cdot\sqrt{\sum_{k=1}^{n}a_k^2})>c$
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$\begingroup$ Such problems are known as "small ball probabilities". Checkout Theorem 8.1 of this manuscript by Rudelson et al. www-personal.umich.edu/~rudelson/papers/rv-smallball.pdf $\endgroup$– dohmatobCommented Jun 24, 2020 at 14:37
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$\begingroup$ Also related are chi-squared variables[1] and their tail inequalities[2]. [1] en.wikipedia.org/wiki/Chi-square_distribution [2] stats.stackexchange.com/questions/4816/… $\endgroup$– usulCommented Jun 25, 2020 at 14:02
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